I don’t think the differences between Old, Classical, Modern Latin are important for something first learning the language. Moreover these categories are fairly artificial and, for the most part, don’t reflect how the language is actually used. In this respect, Latin isn’t like Greek, where the differences between modern and classical Greek are substantial.
As Hampie says, what is called “Old” Latin is very uncommon and more or less only an interest for linguists who study the development of the language.
If you are thinking of famous authors like Cicero, Virgil, Ovid, etc., you’re thinking of what’s called “Classical Latin.” This is the form of the language taught everywhere, and it’s what almost everyone is interested in learning. Sometimes you’ll hear about “Golden Age” Latin, in comparison with the later “Silver Age.” This refers only to different literary styles and tastes, not the language itself.
As far as almost everyone is concerned, Classical Latin is the same as Modern Latin. Strictly speaking, modern Latin is only “modern” because it was written after ancient Roman times–the language itself didn’t really change. During the medieval ages the Latin shows a variety of changes (often in terms of spelling and writing), but this is more idiosyncratic than a “new” stage of the language. Later, during the Renaissance, scholars strove to cultivate the classical style.
What this means for you: study from whatever textbook you find that looks useful/interesting. All textbooks will teach you Classical Latin. If you learn the standard form of the language, you can read anything.
Addendum: If you are in fact only interested in reading Latin texts written in the medieval age, or by scientists, or some other very specific type of texts, there may be reason to skip the traditional textbooks. However, this applies to almost nobody (usually just scholars who are using specific sources for their research projects).
